King Fahd University of
Petroleum and Minerals
Department of Mathematics and Statistics
Workshop on Fractional Models in Science & Engineering (FMSE18) Theory and Computation Monday, December 10, 2018 KFUPM Conference Room (Building # 20) |

Middle Tennessee State University, USA.

Email: Abdul.Khaliq@mtsu.edu

A split-step second order predictor-corrector method
for space-fractional reaction-diffusion equations with nonhomogeneous
boundary conditions is presented and Matrix Transfer Technique (MTT) is used
for spatial discretization. The method is shown to be unconditionally stable
and second order convergent. Numerical experiments are performed to confirm
the stability and second order convergence of the method. The split-step
predictor-corrector method is also compared with an IMEX predictor-corrector
method which is found to incur oscillatory behavior for some time steps. Our
method is seen to produce reliable and oscillation free results for any time
step when implemented on numerical examples with no smooth initial data. We
also present a priori reliability constraint for IMEX predictor-corrector
method to avoid unwanted oscillations and show its validity numerically on
several test problems from the literature.

University of Bari, Italy.

Email:
roberto.garrappa@uniba.it

**On special fractional operators in
computational electromagnetism: theory and numerical methods.**

The propagation of electric and magnetic fields
in complex systems, such as for instance special polymers or biologic
tissues, is usually described in the frequency domains by means of functions
with more than one real powers. The characterization of these models in the
time domain involves new non-standard differential operators of fractional
order [1]. In this talk we discuss the constitutive law of the
Havriliak-Negami model which allows to close the Maxwell's system and
describe the interactions between electric and magnetic fields in a wide
range of materials, with important applications in engineering and medicine. After
describing new recent developments for characterizing in the time domain the
operators associated with this model, we focus with the connections with
fractional integrals and fractional derivatives based on the so-called
Prabhakar function [2] and we discuss some of the possible approaches for
the discretization of these operators and for their use in numerical
simulations [3].

[1] R. Garrappa, F. Mainardi, G. Maione, Models of dielectric relaxation based on completely monotone

functions. Fract. Calc. Appl. Anal., 2016, 19(5), 1105-1160

[2] R. Garra, R. Garrappa, The
Prabhakar or three parameter Mittag--Leffler function: theory and
application.

Commun. Nonlinear Sci. Numer. Simul., 2018, 56, 314-329

[3] R. Garrappa, On Grunwald-Letnikov operators for fractional relaxation in Havriliak-Negami models, Commun.

Nonlinear Sci. Numer. Simul., 2016, 38, 178-191

[1] R. Garrappa, F. Mainardi, G. Maione, Models of dielectric relaxation based on completely monotone

functions. Fract. Calc. Appl.

Commun. Nonlinear Sci. Numer. Simul., 2018, 56, 314-329

[3] R. Garrappa, On Grunwald-Letnikov operators for fractional relaxation in Havriliak-Negami models, Commun.

Nonlinear Sci. Numer.

Department of Electrical & Computer Engineering

University of Porto, Portugal.

Email: jtm@isep.ipp.pt

**Fractional Calculus: The Perspective of Complex Systems.**

**
Salman Malik**

Department of Mathematics

COMSATS University, Pakistan.

Email: salman.amin.malik@gmail.com

**Inverse
Problems for Some Space-Time Fractional Differential Equations.**

Fractional Calculus (FC), that is, the study of
arbitrary order integrals and derivatives has beenconsidered by many experts
from all fields of sciences. FC has applications in many fields just
tomention a few are in biology, physics, finance, viscoelasticity processes,
prediction of extreme eventslike earthquake etc. The fractional derivatives
in time and space have been considered to explain theanomalies in the
complex phenomena of diffusion/transport at both micro and macro level. The
timefractional diffusion equations with Riemann-Liouville or Caputo
fractional derivatives are equivalentto infinitesimal generators of time
fractional evolutions that arise in the transition from microscopicto
macroscopic time scales. The transition from first order derivative to the
fractional order timederivative arise physically reported by many see for
example [1].

Inverse problems for fractional differential equations have been considered by many in the recentpast which include inverse source problems, inverse coefficient problems etc. Indeed, the inverse problemsfor time, space and time-space fractional diffusion equations are considered. In this talk weare going to discuss the inverse source problems for the Space-Time Fractional Differential Equations(STFDEs). The STFDE is obtained from classical diffusion equation by replacing time derivativewith fractional order time derivative and Sturm-Liouville operator by fractional order Sturm-Liouvilleoperator. The spectral problem of the STFDEs is the fractional order Sturm-Liouville system, whichis a generalization of the well known Sturm-Liouville system involving fractional derivatives in spacedefined in Caputo’s sense with Dirichlet boundary conditions. By variational techniques several properties of eigenvalues and eigenfunctions of the fractional order Sturm-Liouville system were investigated in [2].

In the first part of my talk, an inverse problem of recovering a space dependent source term along with solution for a STFDE from over-specified condition of final data will be discussed [3].

The conditions on the given data such that a unique solution of the inverse problem exists will be presented. Secondly, inverse problem of determining a time dependent source term from the total energy measurement of the system (the over-specified condition) for a STFDE will be considered [4].

The existence and uniqueness results are proved by using eigenfunction expansion method. Several special cases, particular examples will be provided. Some future perspectives will be shared with the audience.

Inverse problems for fractional differential equations have been considered by many in the recentpast which include inverse source problems, inverse coefficient problems etc. Indeed, the inverse problemsfor time, space and time-space fractional diffusion equations are considered. In this talk weare going to discuss the inverse source problems for the Space-Time Fractional Differential Equations(STFDEs). The STFDE is obtained from classical diffusion equation by replacing time derivativewith fractional order time derivative and Sturm-Liouville operator by fractional order Sturm-Liouvilleoperator. The spectral problem of the STFDEs is the fractional order Sturm-Liouville system, whichis a generalization of the well known Sturm-Liouville system involving fractional derivatives in spacedefined in Caputo’s sense with Dirichlet boundary conditions. By variational techniques several properties of eigenvalues and eigenfunctions of the fractional order Sturm-Liouville system were investigated in [2].

In the first part of my talk, an inverse problem of recovering a space dependent source term along with solution for a STFDE from over-specified condition of final data will be discussed [3].

The conditions on the given data such that a unique solution of the inverse problem exists will be presented. Secondly, inverse problem of determining a time dependent source term from the total energy measurement of the system (the over-specified condition) for a STFDE will be considered [4].

The existence and uniqueness results are proved by using eigenfunction expansion method. Several special cases, particular examples will be provided. Some future perspectives will be shared with the audience.

- R. Hilfer, Applications of Fractional Calculus in Physics, World Scienti c, 2000.
- M. Klimek, A.B.
Malinowska, T. Odzijewicz, Variational methods for the fractional
Sturm-Liouville

problem, J. Math. Anal. Appl., 416, (2014), 402-428. - M. Ali, S. Aziz and S.A.
Malik, Inverse source problem for a space-time fractional diffusion
equation,

Fract. Calc. Appl. Anal., 21 (2018), 844-863. - M. Ali, S. Aziz, S.A.
Malik, Inverse problem for a space-time fractional diffusion equation:
Application

of fractional Sturm-Liouville operator, Mathematical Methods in the Applied Sciences, 41 (2018),

2733-2744.

Saudi Aramco Expec ARC Computational Modeling Technology (CMT) Team

Email: ozgur.kirlangic@aramco.com

Numerical simulation of fractured reservoirs is a challenging task due to the huge difference between the scales of velocity in the fractures and the rock matrix. In fractured reservoirs, the matrix contributes most of the fluid storage but very little on fluid transport; on the contrary, the fracture system provides high flow capacity but low pore volume [1]. A common approach in dealing with this problem is applying dual or multi-porosity models, which is based on separation of scales. This talk will be about the current state of art in the applied fractured reservoir simulation and explore where the fractional diffusion equations would fit and make difference for the future applications.